Scales of Space and Time

One thing many people know about astronomy is that
it deals with unimaginably large distances and absurdly long times.
A simple trick allows astronomers to think about vast amounts of
time and space without getting confused. We will learn this trick
and use it to gain an overview of the size and age of things in the
universe.

Topics

Linear & Logarithmic Scales

Powers of Ten

The Sizes of Things

Cosmic Time

Scientific Notation

Working With Units

Reading

Figure It Out 1.2: Scientific Notation

p. 4

Figure It Out 1.1: Keeping Track of Space and Time

p. 3

A Closer Look 1.1: A Sense of Scale: Measuring Distances

p. 12

1.1 Peering Through the Universe: A Time Machine

p. 2

1.4 How do You Take a Tape Measure to the Stars

p. 11

Linear Scales

An ordinary ruler is a good example of a linear
scale:

Marks are spaced equally along the scale

Numbers increase by a constant AMOUNT
(in this case by 1)

Logarithmic Scales

A logarithmic scale is labeled a bit differently:

Marks are still spaced equally along the scale

Numbers now increase by a constant FACTOR
(in this case a factor of 10)

Question 1.1

Let's extend the logarithmic scale one space to the
right as shown. What is the value of X?

1,000,001

2,000,000

10,000,000

100,000,000

none of the above

Question 1.2

Now, let's extend the logarithmic scale one space
to the left as shown. What is the value of X?

0

0.1

0.2

0.9

none of the above

What's Wrong With Linear Scales?

A linear scale is fine for comparing sizes of
different kinds of fruit:

It's also fine for comparing diameters of different
planets:

But it's useless if you want to compare fruit and
planets on the same scale!

Why Do We Need Logarithmic Scales?

Using a logarithmic scale, we can easily plot fruit
and planets together:

From this chart, we can see at a glance that a grape
is smaller relative to other kinds of fruit than the Moon is
relative to other planets.

Powers of Ten

Counting zeros when working with very large or very
small numbers is tedious and leads to mistakes, so we will use
powers-of-ten notation; for example:

106 = 1,000,000

10-6 = 1 ÷ 1,000,000 = 0.000,001

The general rule is that 10n,
where n is a positive whole number, is the product of
n factors of 10:

10n

=

10 × 10 × . . . × 10

-- n copies of 10 --

while if n is negative then you divide 1 by
the number you would get using the absolute value of n:

Web Resources

An interactive web page illustrating the same idea as the
Powers of Ten movie.

Reading Between the Marks. I

On linear scale it's pretty clear what we mean
when a value is plotted exactly between two numbered marks,
as X is here:

Here X = 3.5; you can compute that by
averaging the marked values:

X = (3 + 4) ÷ 2 = 7 ÷ 2 = 3.5

What about Y, which is exactly between 3
and X? All you need to do is average those values:

Y = (3 + 3.5) ÷ 2 = 6.5 ÷ 2 = 3.25

You can fill in the rest of the scale by
using this rule over and over.

Reading Between the Marks. II

What about a logarithmic scale? What value does X
have here?

Instead of adding, you multiply, and
instead of dividing by 2, you take the square root:

X = √ (1,000 × 10,000) = √ 10,000,000 = 3,162.3

What about Y, which is exactly between 1000
and X? Apply the same rule again:

Y = √ (1,000 × 3,162.3) = √ 3,162,300 = 1,778.3

Again, you can fill the scale by using this rule
over and over.

The Sizes of Things

We now know enough to chart the sizes of things in
the Powers of Ten movie, using a logarithmic scale with a
factor of 105 between marks:

At a glance this shows us where the physical
scale we're familiar with fits into the Universe as whole; we're
about 1021 times smaller than galaxies, and
109 times smaller than stars, but 1010 times
bigger than atoms.

Question 1.3

Using the chart above, estimate how many times
bigger galaxies are than stars.

106

109

1012

1015

1018

Cosmic Time. I

The book uses this chart to illustrate the history
of the Universe.

This is a linear time scale, so it's hard to see
where human beings fit in.

Cosmic Time. II

We can use a logarithmic scale to show how human
time scales compare to cosmic time.

Actually, this chart, like the one before, is more
about the history of our planet than the history of the universe
as a whole. To really appreciate the history of the universe, we
need to chart time since the Big Bang.

Excerpt from Old Woodrat's Stinky House

Three hundred something million years
the solar system swings around
with all the Milky Way -

Ice ages come one hundred fifty million years apart
last about ten million
then warmer days return -

A venerable desert woodrat nest of twigs and shreds
plastered down with ambered urine
a family house in use eight thousand years,
& four thousand years of using writing equals
the life of a bristlecone pine -

A spoken language works
for about five centuries,
lifespan of a douglas fir;
big floods, big fires, every couple hundred years,
a human life lasts eighty,
a generation twenty.

Hot summers every eight or ten,
four seasons every year
twenty-eight days for the moon
day/night the twenty-four hours

& a song might last four minutes,

a breath is a breath.

Scientific Notation

We use scientific notation to help with the
arithmetic of large and small numbers.

1,230,000 = 1.23 × 106

0.00000123 = 1.23 × 10-6

The same number can take different forms:

1,230,000 = 1.23 × 106 =
12.3 × 105 = 0.123 ×
107

The form 1.23 × 106 is usually preferred,
because the constant in front (1.23) is between 1 and 10.

To multiply, you add the exponents:

(1.2 × 106) × (2
× 105) = (1.2 × 2) ×
10(6+5) = 2.4 × 1011

To divide, you subtract the exponents:

(4.2 × 1012) ÷ (2
× 108) = (4.2 ÷ 2) ×
10(12-8) = 2.1 × 104

To add or subtract numbers in scientific notation,
you have to make the exponents the same first:

(1.2 × 106) + (2 ×
105) = (1.2 × 106) + (0.2 ×
106) = 1.4 × 106

Question 1.4

A neutron star contains as many atoms as an
ordinary star, but each atom has been squashed by gravity to the
size of its central nucleus. Using the chart above, estimate the
diameter of a neutron star.

103 meters

104 meters

105 meters

106 meters

107 meters

Working With Units

Units are a valuable tool; careful attention to the
units at each step of a calculation can help you fix mistakes. The
idea is to work with units as if they were symbols like those in
algebra. For example:

Multiply units along with numbers:

(5 m) × (2 sec) = (5
× 2) × (m × sec) = 10 msec.

The units in this example are meters times
seconds, pronounced `meter seconds' and written `msec'.

Divide units along with numbers:

(10 m) ÷ (5 sec) = (10
÷ 5) × (m ÷ sec) = 2
m/sec.

The units in this example are meters divided
by seconds, pronounced `meters per second' and written
`m/sec'; these are units of
speed.

Cancel when you have the same units on top and bottom:

(15 m) ÷ (5 m) = (15
÷ 5) × (m ÷ m) = 3.

Here the result has no units of any
kind! This is a `pure' number. It has the same value
no matter what units were used for the measurements.

To add or subtract, convert both numbers to the
same units first:

(5 m) + (2 cm) = (5 m)
+ (0.02 m) = (5 + 0.02) m = 5.02 m.

Recall that 1 cm = 0.01 m, so
2 cm = 0.02 m.

You can't add or subtract two numbers unless you can
convert them both to the same units:

(5 m) + (2 sec) = ???

Meters and seconds are different kinds
of quantities; one is length, and the other is time. We
can't convert one to the other, so there is no way to add
them.

Converting between different units is not hard if
you remember to treat units like symbols; replace the original unit
with its equivalent in the unit desired, and do the necessary
arithmetic. For example: